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مرکز یادگیری / جبر

Rational expressions: simplify, multiply, divide, add and subtract

What rational expressions are, their excluded values, and how to simplify, multiply, divide, add and subtract them — then solve rational equations.

2 دقیقه مطالعهبه‌روزرسانی ۱۱ آبان ۱۴۰۵نوشته Sumzo teamبازبینی: Sumzo team

Rational expressions are fractions whose top and bottom are polynomials, such as x2−9x+3\frac{x^2 - 9}{x + 3} or 2x−1\frac{2}{x - 1}. They follow the same rules as number fractions — simplify by cancelling common factors, multiply straight across, add over a common denominator — with one extra rule: the denominator can never be zero.

Excluded values

An excluded value is a value of xx that makes the denominator zero. For x+1x2−4\frac{x + 1}{x^2 - 4}, the denominator x2−4=(x−2)(x+2)x^2 - 4 = (x - 2)(x + 2) is zero at x=2x = 2 and x=−2x = -2, so both are excluded. Find them before you simplify — they stay excluded even if the factor cancels.

How to simplify rational expressions

مثال حل‌شده

Easy: factor and cancel

Simplify x2−9x2+5x+6\frac{x^2 - 9}{x^2 + 5x + 6}.

نمایش راه‌حل
  1. Factor the top and bottom.
    (x−3)(x+3)(x+2)(x+3)\frac{(x - 3)(x + 3)}{(x + 2)(x + 3)}
  2. Cancel the common factor (x+3)(x + 3).
    x−3x+2,x≠−3,−2\frac{x - 3}{x + 2}, \quad x \neq -3, -2

پاسخ: x−3x+2\frac{x - 3}{x + 2}, with x≠−3x \neq -3 and x≠−2x \neq -2

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Multiplying and dividing rational expressions

To multiply, factor everything, multiply the numerators and the denominators, and cancel. To divide, multiply by the reciprocal of the second fraction.

مثال حل‌شده

Medium: dividing

Simplify x2−4x+3÷x−2x2+3x\frac{x^2 - 4}{x + 3} \div \frac{x - 2}{x^2 + 3x}.

نمایش راه‌حل
  1. Multiply by the reciprocal.
    x2−4x+3×x2+3xx−2\frac{x^2 - 4}{x + 3} \times \frac{x^2 + 3x}{x - 2}
  2. Factor.
    (x−2)(x+2)x+3×x(x+3)x−2\frac{(x - 2)(x + 2)}{x + 3} \times \frac{x(x + 3)}{x - 2}
  3. Cancel (x−2)(x - 2) and (x+3)(x + 3).
    x(x+2)x(x + 2)

پاسخ: x(x+2)x(x + 2), with x≠−3x \neq -3, x≠0x \neq 0 and x≠2x \neq 2

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Adding and subtracting rational expressions

مثال حل‌شده

Medium: adding with different denominators

Add 1x+1+2x−1\frac{1}{x + 1} + \frac{2}{x - 1}.

نمایش راه‌حل
  1. The common denominator is (x+1)(x−1)(x + 1)(x - 1).
    1(x−1)+2(x+1)(x+1)(x−1)\frac{1(x - 1) + 2(x + 1)}{(x + 1)(x - 1)}
  2. Simplify the numerator.
    3x+1(x+1)(x−1)\frac{3x + 1}{(x + 1)(x - 1)}

پاسخ: 3x+1(x+1)(x−1)\frac{3x + 1}{(x + 1)(x - 1)}

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How to solve rational expressions in equations

An expression is simplified; an equation is solved. When a rational expression is set equal to something — the same skill as solving rational functions for their zeros — multiply every term by the common denominator to clear the fractions, solve, and reject any answer that is an excluded value.

مثال حل‌شده

Hard: an equation with fractions

Solve 2x−1=3x+2\frac{2}{x - 1} = \frac{3}{x + 2}.

نمایش راه‌حل
  1. Excluded values: x≠1x \neq 1 and x≠−2x \neq -2.
  2. Multiply both sides by (x−1)(x+2)(x - 1)(x + 2).
    2(x+2)=3(x−1)2(x + 2) = 3(x - 1)
  3. Solve.
    2x+4=3x−3⇒x=72x + 4 = 3x - 3 \Rightarrow x = 7
  4. Check: 26=39=13\frac{2}{6} = \frac{3}{9} = \frac{1}{3} ✓, and 7 is not excluded.

پاسخ: x=7x = 7

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Common mistakes

  • Cancelling terms instead of factors: in x+3x+5\frac{x + 3}{x + 5} nothing cancels.
  • Adding the denominators: 1x+1y=x+yxy\frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy}, not 2x+y\frac{2}{x + y}.
  • Forgetting excluded values once a factor has cancelled.
  • Keeping a solution of a rational equation that makes a denominator zero.

پرسش‌های متداول

What is a rational expression?

A fraction with a polynomial in the numerator and a non-zero polynomial in the denominator, such as x+1x2−4\frac{x + 1}{x^2 - 4}.

How do you simplify rational expressions?

Factor the numerator and denominator completely, cancel factors that appear in both, and state the excluded values from the original denominator.

How do I solve rational expressions?

An expression on its own is simplified, not solved. If it is part of an equation, clear the fractions by multiplying by the common denominator, solve, and check that no answer is an excluded value.

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